An asymptotic Filon-type method for infinite range highly oscillatory integrals with exponential kernel

In this paper, we present a new quadrature method for semi-infinite range highly oscillatory integrals with integrands of the form f(x)exp[i omega g(x)], where the phase function g and its derivative are positive, unboundedly increasing on a subinterval [c, infinity] of the integration interval. The method is based on approximating f/g' by a linear combination of negative rational powers of the phase function so that the moments can be expressed by the extended exponential integral function. If the magnitude of omega g'(c) is sufficiently large, our method is very efficient in obtaining very high precision approximations to the integral, without computation of derivatives or the inverse of the phase function. The effectiveness of the method is discussed in the light of a set of test examples including the first problem of the SIAM 100-Digit Challenge, the Bessoid integral, and two finite range integrals. We also present a MATHEMATICA program to be used for the implementation of the method. (C) 2012 IMACS. Published by Elsevier B.V. All rights reserved.